We study (s,t)-cuts of second minimum capacity and present the following algorithmic and graph-theoretic results. 1. Vazirani and Yannakakis [ICALP 1992] designed the first algorithm for computing an (s,t)-cut of second minimum capacity using $O(n^2)$ maximum (s,t)-flow computations. For directed integer-weighted graphs, we significantly improve this bound by designing an algorithm that computes an $(s,t)$-cut of second minimum capacity using $O(\sqrt{n})$ maximum (s,t)-flow computations w.h.p. To achieve this result, a close relationship of independent interest is established between $(s,t)$-cuts of second minimum capacity and global mincuts in directed weighted graphs. 2. Minimum+1 (s,t)-cuts have been studied quite well recently [Baswana, Bhanja, and Pandey, ICALP 2022], which is a special case of second (s,t)-mincut. (a) For directed multi-graphs, we design an algorithm that, given any maximum (s,t)-flow, computes a minimum+1 (s,t)-cut, if it exists, in $O(m)$ time. (b) The existing structures for storing and characterizing all minimum+1 (s,t)-cuts occupy $O(mn)$ space. For undirected multi-graphs, we design a DAG occupying only $O(m)$ space that stores and characterizes all minimum+1 (s,t)-cuts. 3. The study of minimum+1 (s,t)-cuts often turns out to be useful in designing dual edge sensitivity oracles -- a compact data structure for efficiently reporting an (s,t)-mincut after insertion/failure of any given pair of query edges. It has been shown recently [Bhanja, ICALP 2025] that any dual edge sensitivity oracle for (s,t)-mincut in undirected multi-graphs must occupy $Ω(n^2)$ space in the worst-case, irrespective of the query time. For simple graphs, we break this quadratic barrier while achieving a non-trivial query time.